Simplified order 4.0 weak Taylor schemes for additive noise

Following Kloeden and Platen [P.E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer-Verlag, Berlin, 1992] Taylor schemes are considered here as the starting point to obtain simplified Taylor schemes replacing the multiple integrals by simpler variables. The conditions that a group of variables has to fulfill so that the new scheme reaches weak-order 4.0 in the additive noise case are given explicitly, as well as the way to find such groups. For a particular selection, a pair of stochastic schemes with order 4.0 in the weak sense, correcting the one proposed in [P.E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer-Verlag, Berlin, 1992], is obtained.